Navier-Stokes systems with quasimonotone viscosity tensor.

NAVIER-STOKES SYSTEMS WITH QUASIMONOTONE VISCOSITY TENSOR

Pierre Dreyfuss, Norbert Hungerb¨uhler²

1Mathematics Department, University of Fribourg, P´erolles, CH-1700 Fribourg, SWITZERLAND

e-mail: norbert.hungerbuehler@unifr.ch

2Mathematics Department, University of Fribourg, P´erolles, CH-1700 Fribourg, SWITZERLAND

e-mail: pierre.dreyfuss@unifr.ch

Abstract: In [1] we investigated a class of Navier-Stokes systems which is motivated by models for electrorheological fluids. We obtained an existence result for a weak solution under mild monotonicity assumptions for the viscos- ity tensor. In this article, we continue the analysis of such systems, but with various notions of quasimonotonicity instead of classical pointwise monotonic- ity assumptions. Moreover we allow the external force to be of a more general form.

AMS Subj. Classification: 35Q30, 76D05

Key Words: Navier-Stokes systems, weak and quasimonotonicity conditions, electrorheological fluids

1 Introduction

1.1 Retrospect of former results

In this paragraph we introduce some notations, and we recall the main results established in [1] in order to relate them later on with the new results which we derive below.

Let Ω⊂IRⁿbe a bounded open domain with Lipschitz boundary. In [1] we considered the following Navier-Stokes system for the velocityu: Ω×[0, T)→ IRⁿ and the pressure P : Ω×[0, T)→IR

This work is supported by the Swiss National Science Foundation under the Grant Number 200020-100051/1

∂u

∂t −divσ(x, t, u, Du) + (u· ∇)u=f−gradP on Ω×(0, T) (1)

divu= 0 on Ω×(0, T) (2)

u= 0 on ∂Ω×(0, T) (3)

u(·,0) =u₀ on Ω (4)

Here, f ∈L^p0(0, T;V⁰) for somep∈[1 +_n+2²ⁿ ,∞), whereV consists of all func- tions inW₀^1,p(Ω,IRⁿ) with vanishing divergence. Moreoveru₀ ∈L²(Ω; IRⁿ) is an arbitrary initial condition satisfying divu₀ = 0, and σ satisfies the conditions (NS0)–(NS2) below. We allow the viscosity tensor σ to depend (non-linearly) on x, t, u and Du.

The problem (1)–(4) with theu-dependence of σis motivated by the study of electrorheological fluid flows, as explained in the introduction of [1].

To fix some notation, let IIM^m×n denote the real vector space of m ×n matrices equipped with the inner product M : N = M_ijN_ij (with the usual summation convention).

The following main assumptions are imposed on the viscosity tensor σ:

(NS0) (Continuity) σ : Ω×(0, T)×IRⁿ×IIM^n×n → IIM^n×n is a Carath´eodory function, i.e. (x, t)7→σ(x, t, u, F) is measurable for every (u, F)∈IRⁿ× IIM^n×n and (u, F) 7→ σ(x, t, u, F) is continuous for almost every (x, t) ∈ Ω×(0, T).

(NS1) (Growth and coercivity) There existc₁ >0,c₂ >0,λ₁ ∈L^p0(Ω×(0, T)), λ2 ∈L¹(Ω×(0, T)),λ3 ∈L^(p/α)0(Ω×(0, T)), 0< α < p, such that

|σ(x, t, u, F)| 6 λ₁(x, t) +c₁(|u|^p−1 +|F|^p−1) σ(x, t, u, F) :F > −λ₂(x, t)−λ₃(x, t)|u|^α+c₂|F|^p (NS2) (Monotonicity) σ satisfies one of the following conditions:

(a) For all (x, t)∈Ω×(0, T) and all u∈IRⁿ, the map F 7→σ(x, t, u, F) is a C¹-function and is monotone, i.e.

(σ(x, t, u, F)−σ(x, t, u, G)) : (F −G)>0 for all (x, t)∈Ω×(0, T),u∈IR^m and F, G∈IIM^n×n.

(b) There exists a function W : Ω×(0, T)×IRⁿ×IIM^n×n → IR such that σ(x, t, u, F) = ^∂W_∂F (x, t, u, F), and F 7→W(x, t, u, F) is convex and C¹ for all (x, t)∈Ω×(0, T) and all u∈IRⁿ.

(c) σ is strictly monotone, i.e. σ is monotone and (σ(x, t, u, F) − σ(x, t, u, G)) : (F −G) = 0 implies F =G.

We recall that the main point is that, in (a) and (b), it is not required thatσ is strictly monotone or monotone in the variables (u, F) as it is usually assumed in previous work.

We will work with the following function spaces: Let V :={ϕ ∈C₀^∞(Ω; IRⁿ) : divϕ= 0}.

Then,V denotes the closure ofV in the spaceW^1,p(Ω; IRⁿ). A classical result of de Rham shows, that this space is

V ={ϕ ∈W₀^1,p(Ω; IRⁿ) : divϕ= 0}.

In addition, we will have to work with W^s,2(Ω; IRⁿ), where s >1 + ⁿ₂. Then, we denote by

V_s := the closure ofV in the space W^s,2(Ω) and

H_q := the closure ofV in the spaceL^q(Ω), and H :=H₂.

Furthermore, let W denote the space defined by

W :={v ∈L^p(0, T;V) :∂_tv ∈L^p0(0, T;V⁰)},

where the integrals are to be understood in the sense of Bochner and the time- derivative means here the vectorial distributional derivative. We recall that W is continuously embedded in C⁰([0, T];H) and we always identifyv ∈W with its representative in C⁰([0, T];H).

The main result we have proved in [1] is the following:

Theorem 1 Assume that σ satisfies the conditions (NS0)–(NS2) for some p ∈ [1 + _n+2²ⁿ ,∞). Then for every f ∈ L^p0(0, T; V⁰) and every u₀ ∈ H, the Navier-Stokes system (1)–(4) has a weak solution (u, P), withu∈ W , in the following sense: For every v ∈L^p(0, T;V) there holds

Z T

0

h∂_tu, vidt+ Z T

0

Z

Ω

σ(x, t, u, Du) :Dv dx dt+

+ Z T

0

Z

Ω

(u· ∇)u·v dx dt= Z T

0

hf, vidt, u(0,·) = u₀.

The weak solution in Theorem 1 is more than a classical weak solution and in particular the energy equality is satisfied (see Remark in [1, p. 245] for more details).

For the proof we used a Faedo-Galerkin method. By using the assumptions in (NS1) we constructed a Galerkin sequence (u_m) of approximating solutions.

Several compactness properties were then established in [1] which allowed to extract a subsequence um converging weakly to some u in L^p(0, T;V). But then, the monotonicity assumptions (NS2) (a) or (b) do not allow to use the classical monotonicity method (like in [9]) in order to pass to the limit in the Galerkin equations. To overcome this difficulty we then used a refinement of the monotone operator method (inspired by [3]) which involves the theory of Young measures. To this end, we studied the Young measure ν_(x,t) generated by the sequence of gradients (Dum), and obtained a div-curl inequality which was the key ingredient to pass to the limit in the Galerkin equation. This inequality was formulated as follows:

Lemma 2 (A div-curl inequality) The Young measureν_(x,t) generated by the gradients Du_m of the Galerkin approximations u_m has the property, that for all s ∈[0, T]:

Z s

0

Z

Ω

Z

IIM^n×n

σ(x, t, u, λ)−σ(x, t, u, Du)

: λ−Du

dν(x,t)(λ)dxdt60. (5) In a final step we have shown that the existence result in Theorem 1 follows from the div-curl inequality whenever we use one of the monotonicity condi- tions in (NS2). Moreover we have derived some additional properties of the Galerkin approximations: In case (NS2) (a) there holds σ(x, t, um, Dum) * σ(x, t, u, Du) in L^p0(Ω×(0, T)) (for a subsequence), in case (b) we have in addition σ(x, t, u_m, Du_m)→σ(x, t, u, Du) inL^β(Ω×(0, T)), for allβ ∈[1, p⁰), and in case (c), we even have Dum →Duin L^α(Ω×(0, T)) for all α∈[1, p).

1.2 Extension of the results to quasimonotone viscosity tensors In this paper we intend to consider quasimonotonicity assumptions forσrather than the classical pointwise monotonicity: Instead of (NS2) (a), (b) or (c) which are pointwise monotonicity conditions, we consider the assumptions (NS2) (d) and (e) below which represent monotonicity in an integrated form.

Moreover we consider equation (1) with a source term which is allowed to be of a more general form. More precisely, we replace f ∈ L^p0(0, T;V⁰) by f satisfying the assumption (Hf):

(Hf) f : Ω×(0, T)×Rⁿ ×M^n×n → Rⁿ is a Carath´eodory function in the sense (NS0). Moreover we assume that one of the following additional conditions hold:

(i) For a constant β < p−1 and a function λ4 ∈L^p0(Ω×(0, T)) there holds

|f(x, t, u, F)|6λ₄(x, t) +C |u|^β+|F|^β .

(ii) In addition to (i), the function f is independent of the fourth vari- able, or, for a.e. (x, t) ∈ Ω×(0, T) and all u ∈ Rⁿ, the mapping F →f(x, t, u, F) is linear.

We now introduce the definitions of quasimonotonicity which we are going to use:

Definition 3 A functionη: IIM^n×n →IIM^n×nis calledstrictly quasimonotone, if there exist constants c >0and r > 0such that

Z

Ω

(η(Du)−η(Dv)) : (Du−Dv)dx>c Z

Ω

|Du−Dv|^rdx for all u, v ∈W₀^1,p(Ω).

We say that η is strictly p-quasimonotone, if Z

IIM^n×n

(η(λ)−η(¯λ)) : (λ−λ)dν(λ)¯ >0

for all homogeneous W^1,p gradient Young measures ν with center of mass λ¯=hν,idi which are not a single Dirac mass.

Remarks: (a) Note that the notion of p-quasimonotonicity is phrased in terms of gradient Young measures. Notice that although quasimonotonicity is “monotonicity in integrated form”, the gradient of a quasiconvex function is not necessarily strictly p-quasimonotone. A simple example of a strictly p-quasimonotone function is the following: Assume that ηsatisfies the growth condition

|η(F)|6C|F|^p−1 with p > 1 and the structure condition

Z

Ω

(η(F +Dϕ)−η(F)) :Dϕ dx>c Z

Ω

|Dϕ|^pdx

for a constant c > 0 and for all ϕ ∈ C₀^∞(Ω) and all F ∈ IIM^n×n. Then η is strictly p-quasimonotone. This follows easily from the definition if one uses that for every W^1,p gradient Young measure ν there exists a sequence (Dvk) generating ν for which (|Dv_k|^p) is equiintegrable (see [5], [7]).

(b) In [10], Zhang introduced the following notion of quasimonotonicity:

The continuous function η : IIM^N×n → IIM^N×n is quasimonotone (in the sense

of Zhang) if for every F ∈ IIM^N×n, every open subset G of IRⁿ and every ϕ ∈C₀¹(G; IR^N) there holds

Z

G

η(F +Dϕ) :Dϕdx>0.

However, he uses a stronger notion of quasimonotonicity to prove his results, namely, that

Z

G

η(F +Dϕ) :Dϕdx>c Z

G

|Dϕ|^pdx (6) for a fixed constant c >0, along with the growth condition

|η(F)|6C|F|^p−1 (7) for some constantC > 0. We would like to indicate that our definition of strict p-quasimonotonicity can be considered as a generalization of Zhang’s notion of quasimonotonicity. In fact, a function which is quasimonotone in the sense of Zhang, i.e. which satisfies (6) and (7), is strictly p-quasimonotone. To see this, we consider a homogeneous W^1,p gradient Young measure ν with center of mass ¯λ:=hν,idiand which is not a single Dirac mass. Then (for some fixed domain G), there exists a sequence (ϕ_k) in C₀^∞(G) such that the sequence of gradients (Dϕ_k) generates ν and such that (|Dϕ_k|) is equiintegrable (see Remark (a)). By (6) there holds

Z

G

η(F +Dϕk) :Dϕkdx>c Z

G

|Dϕk|^pdx. (8) The sequence (|Dϕk|) is equiintegrable and, by (7), (η(F +Dϕk) : Dϕk) is also equiintegrable. Hence by Ball’s fundamental theorem on Young measures (see [2]), we obtain from (8), that

Z

G

Z

IIM^n×n

η(F +λ) :λdν_x(λ)dx>c Z

G

Z

IIM^n×n

|λ|^pdν_x(λ)dx, and since ν, by hypothesis, is homogeneous,

Z

IIM^n×n

η(F +λ) :λdν(λ)>c Z

IIM^n×n

|λ|^pdν(λ).

A substitution λ=λ⁰−λ, and the special choice¯ F = ¯λ yields Z

IIM^n×n

η(λ⁰) : (λ⁰−λ)dν(λ¯ ⁰)>c Z

IIM^n×n

|λ⁰−¯λ|^pdν(λ⁰)>0 (9) since ν is not a single Dirac mass. Moreover,

Z

IIM^n×n

η(¯λ) : (λ⁰−¯λ)dν(λ⁰) =η(¯λ) : Z

IIM^n×n

λ⁰dν(λ⁰)

| {z }

=¯λ

−η(¯λ) : ¯λ Z

IIM^n×n

dν(λ⁰)

| {z }

=1

= 0.

(10)

If we subtract (10) from (9), we arrive at Z

IIM^n×n

(η(λ⁰)−η(¯λ)) : (λ⁰−λ)dν(λ¯ ⁰)>0,

and hence η is strictly p-quasimonotone. We end this remark by noting that we can replace the power p on the right hand side of (6) by any power r > 0

and still have the same conclusion. ♦

So, in addition to (NS2) (a), (b) and (c), we will now consider the condi- tions:

(NS2) (Monotonicity) σ satisfies one of the following conditions:

(d) for a.e (x, t) ∈ Ω × (0, T) and all u ∈ Rⁿ, the function F → σ(x, t, u, F) is strictly quasimonotone.

(e) for a.e (x, t) ∈ Ω × (0, T) and all u ∈ Rⁿ, the function F → σ(x, t, u, F) is strictlyp-quasimonotone.

The main result we prove in this paper is the following:

Theorem 4 Assume thatσsatisfies the conditions (NS0) and (NS1) for some p∈[1 + _n+2²ⁿ ,∞). Let u₀ be given in H. Then we have:

(i) If σ satisfies one of the condition (NS2) (c), (d) or (e) then for every f satisfying (Hf) (i), the Navier-Stokes system (1)–(4) has a weak solution (u, P), with u∈W .

(ii) If σ satisfies one of the condition (NS2) (a) or (b) then for every f satisfying (Hf) (ii) the same conclusion holds.

Remark: The notion of a solution u ∈ W in Theorem 4 is the same as in Theorem 1: We have u(0,·) =u₀ and there holds

Z T

0

h∂_tu, vidt+ Z T

0

Z

Ω

σ(x, t, u, Du) :Dv dx dt+ Z T

0

Z

Ω

(u· ∇)u·v dx dt=

= Z T

0

Z

Ω

f(x, t, u, Du)·v dx dt ∀v ∈L^p(0, T;V).

1.3 Organization of the paper

In Section 2 we will resume the relevant results presented in [1, Section 1–

6]. We will prove first that for every f satisfying (Hf) (i) we can construct a Galerkin sequence u_m of approximating solutions. In a second step we will show that the convergence properties for um established in [1] remain valid.

We will then recover the same properties for the Young measure associated to (u_m, Du_m), and actually the div-curl inequality holds again.

In Section 3 we pass to the limit m → ∞ in the Galerkin equations and prove Theorem 4. Like in [1], the key ingredient for identifying the weak limit σ(x, t, u_m, Du_m) * σ(x, t, u, Du) (where u is the weak limit in L^p(0, T;V) of a relabeled subsequence of um) is the div-curl lemma (unless for the easier situation when (NS2) (d) holds). In the cases (NS2) (c), (d) or (e) we also get that Du_m → Du in measure, which allows to conclude the first part of Theorem 4. In the situation (NS2) (a) or (b), this additional property of convergence does not hold in general and we have to consider the stronger assumption (Hf) (ii) instead of (Hf) (i) in order to identify the weak limit in f(x, t, um, Dum)* f(x, t, u, Du).

2 The Galerkin approximation 2.1 The Galerkin base

Let the functions w_i ∈ V_s be a Galerkin base, as introduced in [1, Section 2]. We have shown that W := {w1, w2, . . .} is an orthonormal Hilbert base of H. In particular, the L²-orthogonal projector P_m : H → H onto span(w₁, w₂, . . . , w_m),m ∈Nis defined by the formula

P_mu=

m

X

i=1

(w_i, u)_Hw_i. (11)

Of course, the operator norm kPmk_{L (H,H)} = 1. We have also shown that kP_mk_L(Vs,Vs) = 1 and thatP_mconverges pointwise to the identity inL (V_s, V_s).

2.2 The Galerkin approximation

Letm ∈N. In the (Faedo-)Galerkin method one makes an Ansatz for approx- imating solutions u_m of the form

u_m(x, t) =

m

X

i=1

c_mi(t)w_i(x), (12)

where c_mi : [0, T) → IR are supposed to be continuous bounded functions.

We take care of the initial condition (4) by choosing the initial coefficients c_mi:=c_mi(0) = (u₀, w_i)_L² such that

u_m(·,0) =

m

X

i=1

c_miw_i(·)→u₀ in L²(Ω) as m→ ∞. (13)

We try to determine the coefficientscmi(t) in such a way, that for everym ∈N the system of ordinary differential equations

(∂_tu_m, w_j)_H + Z

Ω

σ(x, t, u_m, Du_m) :Dw_jdx+b(u_m, u_m, w_j) =

= Z

Ω

f(x, t, u_m, Du_m)·w_jdx (14) (with j ∈ {1,2, . . . , m}) is satisfied in the sense of distributions. In (14), we used the shorthand notation

b(u, v, w) :=

Z

Ω

((u· ∇)v)·w dx.

Let ε, J, r and K be the quantities introduced in [1, Section 3.1]. For any j = 1, . . . , m we can verify (by using the assumption (Hf)) that the function Θ :J×K →R defined by

Θ(t, c₁, . . . , c_m) :=

Z

Ω

f(x, t,

m

X

i=1

c_iw_i,

m

X

i=1

c_iDw_i)·w_jdx is a Carath´eodory function. Moreover we obtain the estimate:

|Θ(t, c1, . . . , cm)|6C Z

Ω

λ4(x, t)dx+C,

where C may depend on m and r but is independent of t. These results allow to conclude as in [1, Section 3.1] that there exists a local solution for equation (14). This solution um is on the form (12) and it verifies (14) in the sense D ⁰(0, ε⁰), for some ε⁰ >0 which, for the moment, may depend on m.

This local solution can be extended to the whole interval [0, T) independent ofm. To do this one can use the arguments presented in [1, Section 3.2]. More precisely, let τ be arbitrary in the existence interval. We have to replace the term III =Rτ

0hf(t), u_midtbyIII =Rτ 0

R

Ωf(x, t, u_m, Du_m)·u_mdxdt. By using the growth condition (Hf) (i) we then obtain

III 6kλ₄k_Lp0

(Ω×(0,T))ku_mk_L^p(Ω×(0,T))+Cku_mk^β+1_Lp(0,T;V).

This inequality, in combination with the estimates for the terms I andII from [1, Section 3.2] which remain unchanged, permits again to obtain the estimate

|(c_mi(τ))_i=1,...,m|²_IR^m =ku_m(·, τ)k²_L2(Ω)6C¯

for a constant ¯C which is independent of τ (and of m). It follows that the functions c_mj can be extended to the whole interval [0, T] and u_m(x, t) = Pm

j=1c_mj(t)w_j(x) is a solution (not necessarily unique) of (14) in the sense D ⁰(0, T). Moreover we obtain again

ku_mk_C⁰_([0,T];L²_(Ω)) 6C. (15)

2.3 Basic convergence properties

We easily recover the basic convergence properties presented in [1, Section 4.1], i.e. we may extract a subsequence, still denoted by u_m, verifying

u_m * u^∗ in L^∞(0, T;H) (16) u_m * u in L^p(0, T;V) (17)

−divσ(x, t, u_m, Du_m)* χ in L^p0(0, T;V⁰) (18) for some u ∈ L^p(0, T;V)∩L^∞(0, T;H) and χ ∈ L^p0(0, T;V⁰). Moreover, by using (17) together with (Hf) (i) we may also assume that

f(x, t, u_m.Du_m)* ξ in L^p0(Ω×(0, T)) (19) for some ξ ∈L^p0(Ω×(0, T)).

The principal difficulty will be, as in [1], to show thatχ=−divσ(x, t, u, Du).

Moreover here, we will also have to identify ξ with f(x, t, u, Du).

2.4 Convergence in measure

We recall first that for anyqsatisfying 2< q < p^∗ := _n−p^np we have the following chain of continuous injections:

V ,→ⁱ Hq i0

,→H

∼γ

=H⁰ ,→ⁱ¹ V_s⁰. (20) Here, H ∼= H⁰ is the canonical isomorphism γ between the Hilbert space H and its dual. We take over from [1] the notation j ◦i to denote the canonical injection of V into V_s⁰, that is, for u∈V :

hj◦i◦u, vi= Z

Ω

uv dx ∀v ∈V_s.

Next, we consider the time derivative of u_m inD ⁰(0, T;V_s⁰), which is in fact a function in L^p0(0, T;V_s⁰) given by the formula :

h∂_t(j◦i◦u_m), vi= Z

Ω

∂_tu_m(x, t)P_mv(x)dx=− Z

Ω

σ(x, t, u_m, Du_m) :D(P_mv)dx−

−b(u_m, u_m, P_mv) + Z

Ω

f(x, t, u_m, Du_m)P_mvdx ∀v ∈V_s, a.e. t ∈(0, T).

(21) By using the fact that u_m is a bounded sequence inL^p(0, T;V) together with the growth property (Hf) (i) we obtain

Z

Ω

f(x, t, u_m, Du_m).P_mvdx

6Ckvk_Vskf(x, t, u_m, Du_m)k_L¹_(Ω) 6γ_m(t)kvk_{V s},

where (γm) is a bounded sequence in L^p0(0, T).

Consequently we obtain :

|h∂_t(j◦i◦u_m), vi|6C_m(t)kvk_Vs (22) where (C_m) is a bounded sequence in L^p0(0, T).

Remark: The estimate (25) in [1, p. 255] needs to be modified in the following way: C ∈L^p0(0, T) should be replaced by (C_m), a bounded sequence in L^p0(0, T), like here in estimate (22). This modification has no consequence on the results presented as we will see in the sequel.

From (22) we conclude indeed, that {∂_tj◦i◦u_m}_m is a bounded sequence in L^p0(0, T;V_s⁰).

Consequently, by [1, Lemma 2], we may assume (for a further subsequence) u_m →u in L^p(0, T;L^q(Ω)) for all q < p^∗ and in measure on Ω×(0, T).

(23) 2.5 A regularity result for u

The arguments developed in [1, Section 4.3] are easily carried over: It follows that the time derivative ∂_t(j ◦i◦u) is given by the formula

h∂_t(j◦i◦u), vi= Z

Ω

ξvdx− hχ, vi −b(u, u, v), ∀v ∈V_s, a.e. t ∈(0, T). (24) This permits again to obtain the property ∂t(j ◦i◦u) ∈ L^p0(0, T;V⁰), which implies that u∈W .

2.6 The limiting time values for u

We remark first that by the boundedness of the sequence ∂_t(j ◦ i◦ u_m) in L^p0(0, T;V_s⁰) we may extract a subsequence (not relabeled) such that

∂t(j◦i◦um)* ∂t(j ◦i◦u) inL^p0(0, T;V_s⁰). (25) Next, by the same manner as in [1, Section 4.4], we get

u_m(x,0)→u₀(x) =u(x,0) inH, (26) u_m(·, T)* u(·, T), inH. (27) The property (27) can be extended to all t∈[0, T]. Indeed, we have:

Lemma 5 There exists a subsequence of {u_m} (still denoted byu_m) with the property that for every t∈[0, T]

u_m(·, t)* u(·, t) in H. (28)

By (23) the convergence in (28) is actually strong for almost all t∈[0, T].

Proof. We can take over the proof of [1, Lemma 4] with taking into account the remark after the relation (22). The estimate (25) in [1, p. 255] is not true and we have to replace C(t) by C_m(t), where C_m(t) is a bounded sequence in L^p0(0, T). In fact, after this correction the relation (25) in [1] is the same estimate as (22) in this paper. Recall next that we consider p < ∞ and thus the boundedness ofC_m inL^p0(0, T) implies the equiintegrability property.

Consequently the estimate (43) in [1] holds true when we replaceC(t) byCm(t)

and the rest of the proof follows. 2

2.7 The Young measure generated by the Galerkin approximation The sequence (or at least a subsequence) of the gradients Du_m generates a Young measure ν_(x,t), and since u_m converges in measure to u on Ω×(0, T), the sequence (u_m, Du_m) generates the Young measure δ_u(x,t)⊗ν_(x,t) (see, e.g., [6]). Now, we collect some facts about the Young measure ν in the following proposition:

Proposition 6 The Young measureν_(x,t)generated by the sequence {Du_m}_m has the following properties:

(i) ν_(x,t) is a probability measure on IIM^n×nfor almost all (x, t)∈Ω×(0, T).

(ii) ν_(x,t) satisfiesDu(x, t) = hν_(x,t),idi for almost every(x, t)∈Ω×(0, T).

(iii) ν_(x,t) has finite p-th moment for almost all (x, t)∈Ω×(0, T).

(iv) ν_(x,t) is a homogeneous W^1,p gradient Young measure for almost all (x, t)∈Ω×(0, T).

Proof. For (i), (ii) and (iii) see the proof of Proposition 5 in [1].

(iv) We have to show, that {ν_(x,t)}x∈Ω is for almost all t ∈ (0, T) a W^1,p gradient Young measure. To see this, we take a quasiconvex function q on IIM^n×n with q(F)/|F| → 1 as F → ∞. Then, we fix x ∈ Ω, δ ∈ (0,1) and use inequality (1.21) from [8, Lemma 1.6] with u replaced by u_m(x, t), with a:=u(x, t)−Du(x, t)xand withX :=Du(x, t). Furthermore, we chooser >0 such that B_r(x) ⊂ Ω. Observe, that the singular part of the distributional gradient vanishes for u_m and, after integrating the inequality over the time

interval [t0−ε, t0+ε]⊂(0, T), we get Z t0+ε

t0−ε

Z

Br(x)

q(Du_m(y, t))dydt+

+ 1

(1−δ)r Z t0+ε

t0−ε

Z

Br(x)\B_δr(x)

|um(y, t)−u(x, t)−Du(x, t)(y−x)|dydt>

>|B_δr(x)|

Z t0+ε t0−ε

q(Du(x, t))dt.

Letting m tend to infinity in the inequality above, we obtain Z t0+ε

t0−ε

Z

Br(x)

Z

IIM^n×n

q(λ)dν_(y,t)(λ)dydt+

+ 1

(1−δ)r Z t0+ε

t0−ε

Z

Br(x)\B_δr(x)

|u(y, t)−u(x, t) +Du(x, t)(y−x)|dydt>

>|B_δr(x)|

Z t0+ε t0−ε

q(Du(x, t))dt.

Now, we letε →0 andr→0 and use the differentiability properties of Sobolev functions (see, e.g., [4]) and obtain, that for almost all (x, t₀)∈Ω×(0, T)

Z

IIM^n×n

q(λ)dν_(x,t0)(λ)> |B_δr(x)|

|B_r(x)| q(Du(x, t₀)).

Since δ ∈ (0,1) was arbitrary, we conclude that Jensen’s inequality holds true for q and the measure ν_(x,t) for almost all (x, t) ∈ Ω ×(0, T). Using the characterization of W^1,p gradient Young measures of [7] (e.g., in the form of [8, Theorem 8.1]), we conclude that in fact {νx,t}x∈Ω is a W^1,p gradient Young measure on Ω for almost all t∈(0, T). By the localization principle for gradient Young measures, we conclude then, thatν_(x,t) is a homogeneous W^1,p gradient Young measure for almost all (x, t)∈Ω×(0, T). 2

2.8 A Navier-Stokes div-curl inequality

In this section, we prove a Navier-Stokes version of a “div-curl Lemma” which will be the key ingredient to obtain χ =−divσ(x, t, u, Du). The results and the arguments presented in [1, Section 6] can be carried over with only minor modifications which we want to explain in the sequel.

In a first step we note that Z s2

s1

hχ, uidt+1

2ku(·, s₂)k²_H = Z s2

s1

Z

Ω

ξ·udxdt+1

2ku(·, s₁)k²_H, ∀06s₁ 6s₂ 6T.

(29)

This follows in the same way as the energy equality (46) in [1].

Next, we establish the following lemma:

Lemma 7 (A div-curl inequality) The Young measureν_(x,t) generated by the gradients Du_m of the Galerkin approximations u_m has the property that for all s ∈[0, T] :

Z s

0

Z

Ω

Z

IIM^n×n

σ(x, t, u, λ)−σ(x, t, u, Du)

: λ−Du

dν_(x,t)(λ)dxdt60. (30) Proof. We follow the calculations in [1, Section 6.2]. The inequality (48) in [1] can be obtained without any change, and inequality (49) holds true if we replace hf, ui by R

Ωξudx. By using the property (19) together with (23) we then obtain that

m→∞lim Z s

0

Z

Ω

f(x, t, um, Dum)·umdxdt= Z s

0

Z

Ω

ξ·udxdt.

Hence, by using (26) with (28) we obtain lim inf

m→∞

Z s

0

Z

Ω

σ(x, t, u_m, Du_m) :Du_mdxdt 6

Z s

0

Z

Ω

ξ·udxdt−1

2ku(·, s)k²_H +1

2ku(·,0)k²_H.

The rest of the proof is identical to the proof in [1]. 2 Remarks:

(i) In the proof of the div-curl lemma of we do not use any monotonicity assumption.

(ii) An intermediary result is that, for all s∈[0, T], there holds lim inf

m→∞

Z s

0

Z

Ω

(σ(x, t, u_m, Du_m)−σ(x, t, u, Du)) : (Du_m−Du)dxdt60.

To see this, repeat the proof of Lemma 6 in [1] with the modifications indicated above.

3 Passage to the limit

Here we will pass to the limit m → ∞ in the Galerkin equations and prove Theorem 4. The first step is to identify the weak limit σ(x, t, u_m, Du_m) * σ(x, t, u, Du). This will follow from every monotonicity assumption listed in (NS2). In Subsection 3.1, we treat the special cases (NS2) (c), (d) and (e) for which we also obtain the convergenceDu_m→Du in measure. The conclusion is then given in the last subsection.

3.1 The cases (NS2) (c), (d) and (e)

In these three cases we will prove that we may extract a subsequence with the property

Du_m →Du in measure on Ω×(0, T). (31) We consider first the case (NS2) (d). In this situation, elementary arguments are actually sufficient to prove (31), and we do not need the div-curl inequality:

Observe that we have Z T

0

Z

Ω

|Du_m−Du|^rdxdt6

6C

Z T

0

Z

Ω

(σ(x, t, u_m, Du_m)−σ(x, t, u_m, Du)) : (Du_m−Du)dxdt

6C

Z T

0

Z

Ω

(σ(x, t, u_m, Du_m)−σ(x, t, u, Du)) : (Du_m−Du)dxdt+

+C Z T

0

Z

Ω

(σ(x, t, u, Du)−σ(x, t, u_m, Du)) : (Du_m−Du)dxdt.

(32) We remark now that the limit inferior of the first term on the right hand side of (32) is less than or equal to zero (see remark (ii) after lemma 7). The second term vanishes when m tends to infinity because of (23). It follows that

lim inf

m→∞

Z T

0

Z

Ω

|Du_m−Du|^rdxdt= 0, and thus (31) holds for a subsequence.

In case (NS2) (c) we can take over the proof presented in [1]. It remains to consider the case (NS2) (e). We suppose that ν_(x,t) is not a Dirac mass on a set (x, t) ∈ M ⊂ Ω×(0, T) of positive Lebesgue measure |M| > 0. Then, by the strict p-quasimonotonicity of σ(x, t, u,·), and the fact that ν_(x,t) is a homogeneous W^1,p gradient Young measure (see Section 2.7) for almost all (x, t)∈Ω×(0, T), we have for a.e. (x, t)∈M

Z

IIM^n×n

σ(x, t, u, λ) :λdν_(x,t)(λ)>

>

Z

IIM^n×n

σ(x, t, u, λ)dν_(x,t)(λ) : Z

IIM^n×n

λdν_(x,t)(λ)

| {z }

=Du(x, t) .

Hence, by integrating over Ω×(0, T), we get together with Lemma 7 Z T

0

Z

Ω

Z

IIM^n×n

σ(x, t, u, λ)dν_(x,t)(λ) :Du(x, t)dxdt>

>

Z T

0

Z

Ω

Z

IIM^n×n

σ(x, t, u, λ) :λdν_(x,t)(λ)dxdt >

>

Z T

0

Z

Ω

Z

IIM^n×n

σ(x, t, u, λ)dν_(x,t)(λ) :Du(x, t)dxdt which is a contradiction. Hence, we have ν_(x,t) = δ_Du(x,t) for almost every (x, t) ∈ Ω×(0, T). From this, it follows that Dum → Du on Ω ×(0, T) in measure for m → ∞ (see, e.g., [6]).

3.2 Conclusion For the cases (NS2) (c), (d) and (e) there holds

σ(x, t, u_m, Du_m)→σ(x, t, u, Du) inL^β(Ω×(0, T)),∀β ∈[1, p⁰), (33) f(x, t, u_m, Du_m)→f(x, t, u, Du) inL^β(Ω×(0, T)),∀β∈[1, p⁰). (34) To see this, just use (31), the boundedness of the sequences σ(x, t, u_m, Du_m) and f(x, t, u_m, Du_m) inL^p0(Ω×(0, T)), and apply the Vitali convergence the- orem. It then follows that

−divσ(x, t, u_m, Du_m)* χ=−divσ(x, t, u, Du) inL^p0(0, T;V⁰), (35) f(x, t, u_m.Du_m)* ξ =f(x, t, u, Du) inL^p0(Ω×(0, T)), (36) These properties are sufficient to pass to the limit in the Galerkin equations and to conclude the proof of Theorem 4 in case (i) (see [1, p. 266]).

For the remaining cases (NS2) (a) and (b) the property (31) does not hold in general, but we however obtain σ(x, t, u_m, Du_m) * σ(x, t, u, Du) in L^p0(Ω×(0, T)) by using Lemma 7 in [1, Section 7]. This suffices to conclude the proof of Theorem 4 in case (ii): In fact, in this situation we have assumed that f satisfies the assumption (Hf) (ii), and it remains to show that the property (36) still holds without (31). By using (23) we easily verify that it is true for the particular situation in (Hf) (ii), when f is independent of the fourth variable. In the other situation we have that, for a.e (x, t)∈Ω×(0, T) and all u ∈ Rⁿ, the mapping F → f(x, t, u, F) is linear. Here we argue as follows to identify the weak limit ξ in (36):

f(x, t, u_m, Du_m)* Z

IIM^n×n

f(x, t, u, λ)dν_(x,t)(λ)

=f(x, t, u,·)◦ Z

IIM^n×n

λdν_(x,t)(λ) =f(x, t, u, Du),

where for the last equality we have used the property (ii) of Proposition 6.

We can now again pass to the limit in the Galerkin equations and conclude the proof of Theorem 4 in case (ii). We end this discussion by noting, that the energy equality (29) holds true with ξ replaced by f(x, t, u, Du) and with χ replaced by −divσ(x, t, u, Du).

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